Can I host my own web-site?
Of course you can - if you have the technical knowledge and infrastructure to do so.
The chances are, though (particularly given that you're asking the question), that you won't have the technical knowledge to maintain a server, with all the security and maintenance that is required. It's also costly, not just for the server machine, but also to lease out a fixed i.p address.
Remember, your website should be reachable 24/7, and that depends upon your server. Also remember that there are people out there who spend their time, be it for kicks or crime, trying to screw around with servers and the websites hosted on them.
Presuming that you run a small business/organisation, the chances are that setting up a web server will cost you significantly more in terms of time and money than choosing a web hosting account with a professional company.
If you're convinced about setting up your own server, check out the documentation web hosting Faqs, and choose a company and hosting plan that suits your needs.